We prove that every topologically transitive map f on the real line must satisfy the following properties:(1) The set C of critical points is unbounded.(2) The set f (C) of critical values is also unbounded.(3) Apart from the empty set and the whole set, there can be at most one open invariant set.(4) With a single possible exception, for every element x the backward orbit {y ∈ R : f n (y) = x for some n in N} is dense in R.
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